As the drawing shows, the length of a guitar string is L = 0.799 m. The frets ar

Question

As the drawing shows, the length of a guitar string is L = 0.799 m. The frets are numbered for convenience. A performer can play a musical scale on a single string because the spacing between the frets is designed according to the following rule: When the string is pushed against any fret j, the fundamental frequency of the shortened string is larger by a factor of the twelfth root of two than it is when the string is pushed against the fret j - 1. Assuming that the tension in the string is the same for any note, find the spacing (a) between fret 1 and fret 0 and (b) between fret 7 and fret 6.

Explanation / Answer

The frequency ratio corresponding to the interval between each consecutive semitone is 2^(1/12).

Since the tension and density of the string is constant, then the wave speed in the

string is constant and the wavelengths for a stretched string is given by: = 2L,

and the frequency ratio can be written as:

f'/f = 2L/(2L-d) = 2^(1/12) = 1.059

2L = 2*0.799 = 1.589 m

1.589/(1.589-d) = 1.059

d = 1.589 - (1.589/1.059) = 0.088 m

b) fret 7 corresponds to 1/3 L = 1/3* 0.799 = 0.266 m

= 2L' = 2*0.266 =0.532 m

2^(1/12) =0.532/(0.532-d)

d = 0.532 - 0.532/1.059 = 0.0296 m

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